Gravitational Backreaction of Matter Inhomogeneities

The non-linearity of Einstein's equations makes it possible for small-scale matter inhomogeneities to affect the Universe at cosmological distances. We study the size of such effects using a simple heuristic model that captures the most important backreaction effect due to nonrelativistc matter, as well as several exact solutions describing inhomogeneous and anisotropic expanding universes. We find that the effects are $O(H^2l^2/c^2)$ or smaller, where $H$ is the Hubble parameter and $l$ the typical size scale of inhomogeneities. For virialized structures this is of order $v^2/c^2$, where $v$ is the characteristic peculiar velocity.